This collection of expository articles by a range of established experts and newer researchers provides an overview of the recent developments in the theory of locally compact groups. It includes introductory articles on totally disconnected locally compact groups, profinite groups, p-adic Lie groups and the metric geometry of locally compact groups. Concrete examples, including groups acting on trees and Neretin groups, are discussed in detail. An outline of the emerging structure theory of locally compact groups beyond the connected case is presented through three complementary approaches: Willis' theory of the scale function, global decompositions by means of subnormal series, and the local approach relying on the structure lattice. An introduction to lattices, invariant random subgroups and L2-invariants, and a brief account of the Burger-Mozes construction of simple lattices are also included. A final chapter collects various problems suggesting future research directions.
Foreword George Willis
1. On the role of totally disconnected groups in the structure of locally compact groups Marc Burger
2. Locally compact groups as metric spaces Romain Tessera
3. A short primer on profinite groups John S. Wilson
4. Lectures on Lie groups over local fields Helge Gloeckner
5. Abstract quotients of profinite groups, after Nikolov and Segal Benjamin Klopsch
6. Automorphism groups of trees: generalities and prescribed local actions Alejandra Garrido, Yair Glasner and Stephan Tornier
7. Simon Smith's construction of an uncountable family of simple, totally disconnected, locally compact groups Colin Reid and George Willis
8. The Neretin groups Lukasz Garncarek and Nir Lazarovich
9. The scale function and tidy subgroups Albrecht Brehm, Maxime Gheysens, Adrien Le Boudec and Rafaela Rollin
10. Contraction groups and the scale Phillip Wesolek
11. The Bader-Shalom normal subgroup theorem Swiatoslaw Gal
12. Burger-Mozes' simple lattices Laurent Bartholdi
13. A lecture on invariant random subgroups Tsachik Gelander
14. L2-Betti number of discrete and non-discrete groups Roman Sauer
15. Minimal normal closed subgroups in compactly generated tdlc groups Thibaut Dumont and Dennis Gulko
16. Elementary totally disconnected locally compact groups, after Wesolek Morgan Cesa and François Le Maître
17. The structure lattice of a totally disconnected locally compact group John S. Wilson
18. The centraliser lattice David Hume and Thierry Stulemeijer
19. On the quasi-isometric classification of locally compact groups Yves de Cornulier
20. Future directions in locally compact groups: a tentative problem list Pierre-Emmanuel Caprace and Nicolas Monod
Index.
